Local dynamics of non-invertible maps near normal surface singularities

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : ( X , x 0 ) → ( X , x 0 ) f\colon (X,x_0)\to (X,x_0) , where X X is a complex surface having x 0 x_0 as a normal singularity. We prove that as long as x 0 x_0 is not a cusp singularity of X X , then it is possible to find arbitrarily high modifications π : X π → ( X , x 0 ) \pi \colon X_\pi \to (X,x_0) such that the dynamics of f f (or more precisely of f N f^N for N N big enough) on X π X_\pi is algebraically stable. This result is proved by understanding the dynamics induced by f f on a space of valuations associated to X X ; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f f on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of f f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.

Simple and Population Local Search Approaches for Portfolio Design Problem

This chapter introduces a local search optimization technique for solving efficiently a ðnancial portfolio design problem that consists of assigning assets to portfolios, allowing a compromise between maximizing gains, and minimizing losses. This practical problem appears usually in ðnancial engineering, such as in the design of CDO-squared portfolios. This problem has been modeled by Flener et al., who proposed an exact method to solve it. It can be formulated as a quadratic program on the (0,1) domain. It is well known that exact solving approaches on difficult and large instances of quadratic integer programs are known inefficient. That is why the authors have adopted local search methods, namely simple local search and population local search. They propose neighborhood and evaluation functions specialized on this problem. To boost the local search process, they propose also a greedy algorithm to start the search with an optimized initial configuration. Experimental results on non-trivial instances of the problem show the effectiveness of the incomplete approach.

A result on the Slope conjectures for 3-string Montesinos knots

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.